Question

One kind of cake requires $200g$ of flour and $25g$ of fat, and another kind of cake requires $100g$ of flour and $50g$ of fat. Find the maximum number of cakes that can be made from $5kg$ of flour and $1kg$ of fat assuming that there is no shortage of the other ingredients used in making the cakes.

Answer 1

Let number of first kind of cake is $X$ and another kind of cake is $Y$.

So, total flour required $=200X+100Yg$

and total fat required $=25X+50Yg$

Since, maximum flour available is $5kg=5000g$

$\therefore 200X+100Y\le 5000$

$\Rightarrow 2X+Y\le 50...\left(1\right)$

Also, maximum fat available is $1kg=1000g$

$\therefore 25X+50Y\le 1000$

$\Rightarrow X+2Y\le 40...\left(2\right)$

Since, quantity of cakes can't be negative.

$\therefore X\ge 0,Y\ge 0...\left(3\right)$

We have to maximize number of cakes that can be made.

So, Objective function is $Z=X+Y$

After plotting all the constraints given by equation (1), (2) and (3), we got the feasible region as shown in the image.

Corner points	Value of $Z=X+Y$
A (0,20)	20
B (20,10)	30 (Maximum)
C (25,0)	25

So, maximum cake that can be made is $30$, where first kind of cake will be 20 and second kind of cake will be 10.